Construction 2.6 ([1]})  Define a homomorphism \(\pi _0 \colon ^U\rightarrow ^W\)  sending a bordism class \([M]\)  to the class of the submanifold \(N \subset \mathbb {C}P^1 \times M\)  dual to \(\overline{\eta }\otimes \det \mathcal {T} M\) . We have \(\det \mathcal {T} N\cong i^*\overline{\eta }\) , where \(i\)  is the embedding \(N \hookrightarrow \mathbb {C}P^1 \times M\) , so \(N\)  has a natural \(c_1\) -spherical stably complex structure.

